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Theorem adjadd 28139
Description: The adjoint of the sum of two operators. Theorem 3.11(iii) of [Beran] p. 106. (Contributed by NM, 22-Feb-2006.) (New usage is discouraged.)
Assertion
Ref Expression
adjadd ((𝑆 ∈ dom adj𝑇 ∈ dom adj) → (adj‘(𝑆 +op 𝑇)) = ((adj𝑆) +op (adj𝑇)))

Proof of Theorem adjadd
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmadjop 27934 . . 3 (𝑆 ∈ dom adj𝑆: ℋ⟶ ℋ)
2 dmadjop 27934 . . 3 (𝑇 ∈ dom adj𝑇: ℋ⟶ ℋ)
3 hoaddcl 27804 . . 3 ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆 +op 𝑇): ℋ⟶ ℋ)
41, 2, 3syl2an 492 . 2 ((𝑆 ∈ dom adj𝑇 ∈ dom adj) → (𝑆 +op 𝑇): ℋ⟶ ℋ)
5 dmadjrn 27941 . . . 4 (𝑆 ∈ dom adj → (adj𝑆) ∈ dom adj)
6 dmadjop 27934 . . . 4 ((adj𝑆) ∈ dom adj → (adj𝑆): ℋ⟶ ℋ)
75, 6syl 17 . . 3 (𝑆 ∈ dom adj → (adj𝑆): ℋ⟶ ℋ)
8 dmadjrn 27941 . . . 4 (𝑇 ∈ dom adj → (adj𝑇) ∈ dom adj)
9 dmadjop 27934 . . . 4 ((adj𝑇) ∈ dom adj → (adj𝑇): ℋ⟶ ℋ)
108, 9syl 17 . . 3 (𝑇 ∈ dom adj → (adj𝑇): ℋ⟶ ℋ)
11 hoaddcl 27804 . . 3 (((adj𝑆): ℋ⟶ ℋ ∧ (adj𝑇): ℋ⟶ ℋ) → ((adj𝑆) +op (adj𝑇)): ℋ⟶ ℋ)
127, 10, 11syl2an 492 . 2 ((𝑆 ∈ dom adj𝑇 ∈ dom adj) → ((adj𝑆) +op (adj𝑇)): ℋ⟶ ℋ)
13 adj2 27980 . . . . . . . 8 ((𝑆 ∈ dom adj𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑆𝑥) ·ih 𝑦) = (𝑥 ·ih ((adj𝑆)‘𝑦)))
14133expb 1257 . . . . . . 7 ((𝑆 ∈ dom adj ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑆𝑥) ·ih 𝑦) = (𝑥 ·ih ((adj𝑆)‘𝑦)))
1514adantlr 746 . . . . . 6 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑆𝑥) ·ih 𝑦) = (𝑥 ·ih ((adj𝑆)‘𝑦)))
16 adj2 27980 . . . . . . . 8 ((𝑇 ∈ dom adj𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇𝑥) ·ih 𝑦) = (𝑥 ·ih ((adj𝑇)‘𝑦)))
17163expb 1257 . . . . . . 7 ((𝑇 ∈ dom adj ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇𝑥) ·ih 𝑦) = (𝑥 ·ih ((adj𝑇)‘𝑦)))
1817adantll 745 . . . . . 6 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇𝑥) ·ih 𝑦) = (𝑥 ·ih ((adj𝑇)‘𝑦)))
1915, 18oveq12d 6542 . . . . 5 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑆𝑥) ·ih 𝑦) + ((𝑇𝑥) ·ih 𝑦)) = ((𝑥 ·ih ((adj𝑆)‘𝑦)) + (𝑥 ·ih ((adj𝑇)‘𝑦))))
201ffvelrnda 6249 . . . . . . 7 ((𝑆 ∈ dom adj𝑥 ∈ ℋ) → (𝑆𝑥) ∈ ℋ)
2120ad2ant2r 778 . . . . . 6 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑆𝑥) ∈ ℋ)
222ffvelrnda 6249 . . . . . . 7 ((𝑇 ∈ dom adj𝑥 ∈ ℋ) → (𝑇𝑥) ∈ ℋ)
2322ad2ant2lr 779 . . . . . 6 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑇𝑥) ∈ ℋ)
24 simprr 791 . . . . . 6 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑦 ∈ ℋ)
25 ax-his2 27127 . . . . . 6 (((𝑆𝑥) ∈ ℋ ∧ (𝑇𝑥) ∈ ℋ ∧ 𝑦 ∈ ℋ) → (((𝑆𝑥) + (𝑇𝑥)) ·ih 𝑦) = (((𝑆𝑥) ·ih 𝑦) + ((𝑇𝑥) ·ih 𝑦)))
2621, 23, 24, 25syl3anc 1317 . . . . 5 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑆𝑥) + (𝑇𝑥)) ·ih 𝑦) = (((𝑆𝑥) ·ih 𝑦) + ((𝑇𝑥) ·ih 𝑦)))
27 simprl 789 . . . . . 6 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑥 ∈ ℋ)
28 adjcl 27978 . . . . . . 7 ((𝑆 ∈ dom adj𝑦 ∈ ℋ) → ((adj𝑆)‘𝑦) ∈ ℋ)
2928ad2ant2rl 780 . . . . . 6 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((adj𝑆)‘𝑦) ∈ ℋ)
30 adjcl 27978 . . . . . . 7 ((𝑇 ∈ dom adj𝑦 ∈ ℋ) → ((adj𝑇)‘𝑦) ∈ ℋ)
3130ad2ant2l 777 . . . . . 6 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((adj𝑇)‘𝑦) ∈ ℋ)
32 his7 27134 . . . . . 6 ((𝑥 ∈ ℋ ∧ ((adj𝑆)‘𝑦) ∈ ℋ ∧ ((adj𝑇)‘𝑦) ∈ ℋ) → (𝑥 ·ih (((adj𝑆)‘𝑦) + ((adj𝑇)‘𝑦))) = ((𝑥 ·ih ((adj𝑆)‘𝑦)) + (𝑥 ·ih ((adj𝑇)‘𝑦))))
3327, 29, 31, 32syl3anc 1317 . . . . 5 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih (((adj𝑆)‘𝑦) + ((adj𝑇)‘𝑦))) = ((𝑥 ·ih ((adj𝑆)‘𝑦)) + (𝑥 ·ih ((adj𝑇)‘𝑦))))
3419, 26, 333eqtr4rd 2651 . . . 4 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih (((adj𝑆)‘𝑦) + ((adj𝑇)‘𝑦))) = (((𝑆𝑥) + (𝑇𝑥)) ·ih 𝑦))
357, 10anim12i 587 . . . . . . 7 ((𝑆 ∈ dom adj𝑇 ∈ dom adj) → ((adj𝑆): ℋ⟶ ℋ ∧ (adj𝑇): ℋ⟶ ℋ))
36 hosval 27786 . . . . . . . 8 (((adj𝑆): ℋ⟶ ℋ ∧ (adj𝑇): ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → (((adj𝑆) +op (adj𝑇))‘𝑦) = (((adj𝑆)‘𝑦) + ((adj𝑇)‘𝑦)))
37363expa 1256 . . . . . . 7 ((((adj𝑆): ℋ⟶ ℋ ∧ (adj𝑇): ℋ⟶ ℋ) ∧ 𝑦 ∈ ℋ) → (((adj𝑆) +op (adj𝑇))‘𝑦) = (((adj𝑆)‘𝑦) + ((adj𝑇)‘𝑦)))
3835, 37sylan 486 . . . . . 6 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ 𝑦 ∈ ℋ) → (((adj𝑆) +op (adj𝑇))‘𝑦) = (((adj𝑆)‘𝑦) + ((adj𝑇)‘𝑦)))
3938adantrl 747 . . . . 5 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((adj𝑆) +op (adj𝑇))‘𝑦) = (((adj𝑆)‘𝑦) + ((adj𝑇)‘𝑦)))
4039oveq2d 6540 . . . 4 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih (((adj𝑆) +op (adj𝑇))‘𝑦)) = (𝑥 ·ih (((adj𝑆)‘𝑦) + ((adj𝑇)‘𝑦))))
411, 2anim12i 587 . . . . . . 7 ((𝑆 ∈ dom adj𝑇 ∈ dom adj) → (𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ))
42 hosval 27786 . . . . . . . 8 ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆𝑥) + (𝑇𝑥)))
43423expa 1256 . . . . . . 7 (((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆𝑥) + (𝑇𝑥)))
4441, 43sylan 486 . . . . . 6 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ 𝑥 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆𝑥) + (𝑇𝑥)))
4544adantrr 748 . . . . 5 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆𝑥) + (𝑇𝑥)))
4645oveq1d 6539 . . . 4 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑆 +op 𝑇)‘𝑥) ·ih 𝑦) = (((𝑆𝑥) + (𝑇𝑥)) ·ih 𝑦))
4734, 40, 463eqtr4rd 2651 . . 3 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑆 +op 𝑇)‘𝑥) ·ih 𝑦) = (𝑥 ·ih (((adj𝑆) +op (adj𝑇))‘𝑦)))
4847ralrimivva 2950 . 2 ((𝑆 ∈ dom adj𝑇 ∈ dom adj) → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (((𝑆 +op 𝑇)‘𝑥) ·ih 𝑦) = (𝑥 ·ih (((adj𝑆) +op (adj𝑇))‘𝑦)))
49 adjeq 27981 . 2 (((𝑆 +op 𝑇): ℋ⟶ ℋ ∧ ((adj𝑆) +op (adj𝑇)): ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (((𝑆 +op 𝑇)‘𝑥) ·ih 𝑦) = (𝑥 ·ih (((adj𝑆) +op (adj𝑇))‘𝑦))) → (adj‘(𝑆 +op 𝑇)) = ((adj𝑆) +op (adj𝑇)))
504, 12, 48, 49syl3anc 1317 1 ((𝑆 ∈ dom adj𝑇 ∈ dom adj) → (adj‘(𝑆 +op 𝑇)) = ((adj𝑆) +op (adj𝑇)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  wral 2892  dom cdm 5025  wf 5783  cfv 5787  (class class class)co 6524   + caddc 9792  chil 26963   + cva 26964   ·ih csp 26966   +op chos 26982  adjcado 26999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821  ax-resscn 9846  ax-1cn 9847  ax-icn 9848  ax-addcl 9849  ax-addrcl 9850  ax-mulcl 9851  ax-mulrcl 9852  ax-mulcom 9853  ax-addass 9854  ax-mulass 9855  ax-distr 9856  ax-i2m1 9857  ax-1ne0 9858  ax-1rid 9859  ax-rnegex 9860  ax-rrecex 9861  ax-cnre 9862  ax-pre-lttri 9863  ax-pre-lttrn 9864  ax-pre-ltadd 9865  ax-pre-mulgt0 9866  ax-hilex 27043  ax-hfvadd 27044  ax-hvcom 27045  ax-hvass 27046  ax-hv0cl 27047  ax-hvaddid 27048  ax-hfvmul 27049  ax-hvmulid 27050  ax-hvdistr2 27053  ax-hvmul0 27054  ax-hfi 27123  ax-his1 27126  ax-his2 27127  ax-his3 27128  ax-his4 27129
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-nel 2779  df-ral 2897  df-rex 2898  df-reu 2899  df-rmo 2900  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-op 4128  df-uni 4364  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-id 4940  df-po 4946  df-so 4947  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-riota 6486  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-er 7603  df-map 7720  df-en 7816  df-dom 7817  df-sdom 7818  df-pnf 9929  df-mnf 9930  df-xr 9931  df-ltxr 9932  df-le 9933  df-sub 10116  df-neg 10117  df-div 10531  df-2 10923  df-cj 13630  df-re 13631  df-im 13632  df-hvsub 27015  df-hosum 27776  df-adjh 27895
This theorem is referenced by: (None)
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